Ceres Optimization Problem Construction for Mag-INS Alignment
Published:
The problem to be solved is presented in detail here.
The constrained non-linear least squares problem can be modeled by
\[\begin{aligned} \min_R\quad& \frac{1}{2} \sum_k \rho_k \left( \left\| f_k(R) \right\|^2 \right) \\ \rm{s}.\rm{t}.\quad& R\in\mathbb{SO}(3) \end{aligned}\]where
\[f_k(R)= \boldsymbol{y}_{k+1}^m- \left(\tilde{D}R R^{b_{k+1}}_{b_k} R^T\tilde{D}^{-1}(\boldsymbol{y}_{k}^m-\hat{o}) +\hat{o}\right)\]is the CostFunction that depends on the parameter block parameterizing the rotation matrix \(R\). And \(\rho_k(|f_k(R)|^2)\) is known as a residual block.
\(\rho_k\) is a LossFunction. A LossFunction is a scalar valued function that is used to reduce the influence of outliers on the solution of non-linear least squares problems.